Analysis of Variance
Department of Educational Psychology
Agenda
1 Overview and Introduction
2 Review of One-Sample Cases (Module 9) and Computation
3 Comparing One- and Two-Sample Cases
4 Two Population Means with Unknown Standard Deviations
5 Matched or Paired Samples
6 Conclusion
Agenda
1 Overview and Introduction
2 Review of One-Sample Cases (Module 9) and Computation
3 Comparing One- and Two-Sample Cases
4 Two Population Means with Unknown Standard Deviations
5 Matched or Paired Samples
6 Conclusion
Set up an appropriate alternative hypothesis comparing some variable against some set value, and then write the corresponding null hypothesis.
Set \(\alpha\) at an appropriate level to minimize risk and balance Type I and Type II error chances
Ensure variable is normally distributed (and also, continuous)
Determine whether you have the population standard deviation or not
(Set hypotheses) I predict that professors will spend more than 40 hours this week working on teaching classes
(Set alpha) I decide to set \(\alpha = 0.10\), saying that I am not particularly concerned about Type I errors
(Ensure normality and continuous nature) I’ll assume the hours worked by professor is normally-distributed, and this is a continuous variable because hours worked is continuous
(Do I have population standard deviation?) No, I don’t know what the population standard deviation, thus I know I need to rely on the t-distribution
(Compute p-value to make decision against alpha) I find a p-value of 0.09, which is less than \(\alpha = 0.10\), thus I reject the null hypothesis
(Make decision) I have evidence that professors spend more 40 hours this week working on teaching classes
Agenda
1 Overview and Introduction
2 Review of One-Sample Cases (Module 9) and Computation
3 Comparing One- and Two-Sample Cases
4 Two Population Means with Unknown Standard Deviations
5 Matched or Paired Samples
6 Conclusion
Agenda
1 Overview and Introduction
2 Review of One-Sample Cases (Module 9) and Computation
3 Comparing One- and Two-Sample Cases
4 Two Population Means with Unknown Standard Deviations
5 Matched or Paired Samples
6 Conclusion
\[ SE_{diff} = \sqrt{\frac{(s_1)^2}{n_1} + \frac{(s_2)^2}{n_2}} \]
\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE_{diff}} \]
\[ df = \frac{(\frac{(s_1)^2}{n_1} + \frac{(s_2)^2}{n_2})^2}{(\frac{1}{n_1 - 1})(\frac{(s_1)^2}{n_1})^2 + (\frac{1}{n_2 - 1})(\frac{(s_2)^2}{n_2})^2} \]
Where:
Calculation of p-value is done with calculator or computer
Wait - who’s Welch?
\[ SE_{diff} = \sqrt{\frac{(s_1)^2}{n_1} + \frac{(s_2)^2}{n_2}} \]
\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE_{diff}} \]
\[ df = \frac{(\frac{(s_1)^2}{n_1} + \frac{(s_2)^2}{n_2})^2}{(\frac{1}{n_1 - 1})(\frac{(s_1)^2}{n_1})^2 + (\frac{1}{n_2 - 1})(\frac{(s_2)^2}{n_2})^2} \]
\[ SE_{diff} = \sqrt{\frac{(15)^2}{50} + \frac{(10)^2}{50}} = 2.55 \]
\[ t = \frac{(100 - 120)}{2.55} = -7.84 \]
\[ df = \frac{(\frac{(15)^2}{50} + \frac{(10)^2}{50})^2}{(\frac{1}{50 - 1})(\frac{(15)^2}{50})^2 + (\frac{1}{50 - 1})(\frac{(10)^2}{50})^2} = 84.9 \]
Agenda
1 Overview and Introduction
2 Review of One-Sample Cases (Module 9) and Computation
3 Comparing One- and Two-Sample Cases
4 Two Population Means with Unknown Standard Deviations
5 Matched or Paired Samples
6 Conclusion
\[ \bar{d} = \frac{d_1 + d_2 + ... + d_n}{n} \]
\[ s_d = \frac{\sqrt{(d_1 - \bar{d})^2 + (d_2 - \bar{d})^2 + ... + (d_n - \bar{d})^2}}{n - 1} \]
\[ SE_d = \frac{s_d}{\sqrt{n}} \]
\[ t = \frac{\bar{d}}{SE} \]
\[ df = n - 1 \]
\[ \bar{d} = \frac{d_1 + d_2 + ... + d_n}{n} \]
\[ s_d = \frac{\sqrt{(d_1 - \bar{d})^2 + (d_2 - \bar{d})^2 + ... + (d_n - \bar{d})^2}}{n - 1} \]
\[ SE_d = \frac{s_d}{\sqrt{n}} \]
\[ t = \frac{\bar{d}}{SE} \]
\[ df = n - 1 \]
\[ SE_d = \frac{1.86}{\sqrt{6}} = 0.760 \]
\[ t = \frac{4.67}{0.760} = 6.14 \]
\[ df = 6 - 1 = 5 \]
Agenda
1 Overview and Introduction
2 Review of One-Sample Cases (Module 9) and Computation
3 Comparing One- and Two-Sample Cases
4 Two Population Means with Unknown Standard Deviations
5 Matched or Paired Samples
6 Conclusion
Hypothesis testing with two-samples follows the same process as with one-samples, but introduces comparisons between groups, rather than just testing against some value
When testing two samples, they may be independent or paired, depending on the design of the study, and that changes what computation process we use
The independent-samples test may either assume equal variance’s (Student’s) or not (Welch’s)
While there are two-sample tests for proportions and when the population standard deviation is known, it is far more common to use the t-distribution
Module 10 Lecture - Mixed Effects Designs || Analysis of Variance